Lecture Notes in Differential Equations - Bruce E. Shapiro

118 LESSON 14. REVIEW OF LINEAR ALGEBRA
Definition 14.3. Let v = (x, y, z) and w = (x ′ , y ′ , z ′ ) be Euclidean 3-
vectors. Then the angle between v and w is defined as the angle between
the line segments joining the origin and the points P = (x, y, z) and P ′ =
(x ′ , y ′ , z ′ ).
We can define vector addition or vector subtraction by
v + w = (x, y, z) + (x ′ , y ′ , z ′ ) = (x + x ′ , y + y ′ , z + z ′ ) (14.3)
where v = (x, y, z) and w = (x ′ , y ′ , z ′ ), and scalar multiplcation (multiplication
by a real number) by
kv = (kx, ky, kz) (14.4)
Theorem 14.4. The set of all Euclidean vectors is closed under vector
addition and scalar multiplication.
Definition 14.5. Let v = (x, y, z), w = (x ′ , y ′ , z ′ ) be Euclidean 3-vectors.
Their dot product is defined as
v · w = xx ′ + yy ′ + zz ′ (14.5)
Theorem 14.6. Let θ be the angle between the line segments from the
origin to the points (x, y, z) and (x ′ , y ′ , z ′ ) in Euclidean 3-space. Then
v · w = |v||w| cos θ (14.6)
Definition 14.7. The standard basis vectors for Euclidean 3-space are
the vectors
i =(1, 0, 0) (14.7)
j =(0, 1, 0) (14.8)
k =(0, 0, 1) (14.9)
Theorem 14.8. Let v = (x, y, z) be any Euclidean 3-vector. Then
v = ix + jy + kz (14.10)
Definition 14.9. The vectors v 1 , v 2 , . . . , v n are said to be linearly dependent
if there exist numbers a 1 , a 2 , . . . , a n , not all zero, such that
a 1 v 1 + a 2 v 2 + · · · + a n v n = 0 (14.11)
If no such numbers exist the vectors are said to be linearly independent.